What is the gcd and lcm of 54, 18, 27 and 36

To calculate the **greatest common divisor (GCD)** and **least common multiple (LCM)** of 54, 18, 27, and 36, let's break it down step by step.

### Step 1: Prime Factorization
First, we need the prime factorization of each number.

- **54** = \( 2 \times 3^3 \)
- **18** = \( 2 \times 3^2 \)
- **27** = \( 3^3 \)
- **36** = \( 2^2 \times 3^2 \)

### Step 2: GCD Calculation
The GCD is found by taking the lowest power of all prime factors common to each number.

- For **2**: The lowest power is \( 2^0 \) (as 27 does not have a factor of 2).
- For **3**: The lowest power is \( 3^2 \) (since 18 and 36 have \( 3^2 \)).

Thus, the **GCD** is:
\[
GCD(54, 18, 27, 36) = 3^2 = 9
\]

### Step 3: LCM Calculation
The LCM is found by taking the highest power of all prime factors present in any number.

- For **2**: The highest power is \( 2^2 \) (from 36).
- For **3**: The highest power is \( 3^3 \) (from 54 and 27).

Thus, the **LCM** is:
\[
LCM(54, 18, 27, 36) = 2^2 \times 3^3 = 4 \times 27 = 108
\]

### Final Results:
- **GCD** = 9
- **LCM** = 108

Would you like further clarification or more details?

### 5 Related Questions:
1. How do you find the GCD and LCM of two numbers?
2. What is the significance of the prime factorization method in finding GCD and LCM?
3. Can the GCD of a set of numbers ever be greater than the smallest number in the set?
4. How does the GCD affect the LCM in multiple numbers?
5. Can two numbers have the same GCD but different LCM values?

### Tip:
The product of the GCD and LCM of two numbers is always equal to the product of the numbers themselves!

Learn how to find the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) of 54, 18, 27, and 36 using prime factorization. This step-by-step solution explains the core concepts of number theory and provides detailed calculations. Suitable for students in Grades 6-8.

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